Cron execution VS Execution from GUI

Hello

I wrote a plugin that schedule a WP cron on activation.
The hook calls a function that calls another function included.
The function returns reports values and they are sent via email.

The script works well when I press Execute in the CRON manager admin.
The script works not so well when the cron job is run. The email does not have content of the function. Though I can see with error_log that the function is well executed.

Anyone has a tip on how to investigate further?

Here is the code simplified :

<?php
/*
Plugin Name: MY PLUGIN
*/

require_once(sprintf("%s/folder/import_common_functions.php", dirname(__FILE__)));				

if(!class_exists('WP_My_Plugin'))
{
    class WP_My_Plugin
    {

...
        /**
         * Activate the plugin
         */
        public static function activate()
        {
            // Plan the CRON scheduling
            wp_plugin_schedule();
        }

        /**
         * Deactivate the plugin
         */
        public static function deactivate()
        {
           // Remove CRON when disabling the plugin
           if (  wp_next_scheduled( 'wp_plugin_import' ) ) {
			 		 wp_clear_scheduled_hook( 'wp_plugin_import' );
		}

        } 

			...

    }
} 

if(class_exists('WP_My_Plugin'))
{
...
    // instantiate the plugin class
    $wp_plugin = new WP_My_Plugin();

    function wp_plugin_schedule() {

	    // Scheduling of runiing batch import of films/screenings
	    if (  !wp_next_scheduled( 'wp_plugin_import' ) ) {
				wp_schedule_event( time(), 'hourly', 'wp_plugin_import' );
		}

		}		

		add_action( 'wp_plugin_import', 'scheduled_import' );

		function scheduled_import() {

			$result = null;
			$result = import_function(); // function included in import_common_functions.php

			...

		  $result_mail = wp_mail( $email,
		  'Automatic email from '.get_bloginfo( 'name')." - ".date("Y-m-d h:i"), //email subject
		  $email_content );

			...

		}

}

https://wordpress.org/plugins/advanced-cron-manager/